A System of Nonlinear Operator Equations for a Mixed Family of Fuzzy and Crisp Operators in Probabilistic Normed Spaces
- PDF / 156,397 Bytes
- 17 Pages / 600.05 x 792 pts Page_size
- 51 Downloads / 177 Views
Research Article A System of Nonlinear Operator Equations for a Mixed Family of Fuzzy and Crisp Operators in Probabilistic Normed Spaces Yeol Je Cho,1 Heng-you Lan,2 and Nan-jing Huang3 1
Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-701, South Korea 2 Department of Mathematics, Sichuan University of Science & Engineering, Zigong, Sichuan 643000, China 3 Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China Correspondence should be addressed to Heng-you Lan, [email protected] Received 8 January 2010; Accepted 23 March 2010 Academic Editor: Yuming Xing Copyright q 2010 Yeol Je Cho et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. By using a random version of the theory of contractor introduced by Altman, we introduce and study a system of nonlinear operator equations for a mixed family of fuzzy and crisp operators in probabilistic normed spaces. We construct some new iterative algorithms for solving this kind of nonlinear operator equations. We also prove some new existence theorems of solutions of a new system of nonlinear operator equations for a mixed family of fuzzy and crisp operators and some new convergence results of sequences generated by iterative algorithms under joint orbitally complete conditions.
1. Introduction Altman 1, 2 introduced the theory of contractor and contractor direction, which has a very strong significant for the study of existence and uniqueness for solving nonlinear operator equations in Banach spaces. The theory of contractor offers a unified approach to a very large class of iterative methods including the most important ones. Chang 3 introduced the concept of probabilistic contractor and studied the existence and uniqueness of solution for nonlinear operator equations with probabilistic contractor in Menger PN-spaces. By using the theory of countable extension of t-norms 4–6 and the results from 7, 8, many results for the more general classes of t-norms have been proved see 9 and the references therein. On the other hand, since then, several kinds of variational inequalities, variational inclusions, complementarity problems, and nonlinear equations with fuzzy mappings were
2
Journal of Inequalities and Applications
introduced and studied by many authors see, e.g., 8–15. Sharma et al. 15 considered two nonfuzzy mappings and a sequence of fuzzy mappings to define a hybrid D-compatible condition. They also showed the existence of common fixed points under such condition, where the range of the one of the two nonfuzzy mappings is joint orbitally complete. Furthermore, Cho et al. 10 introduced the concept of probabilistic contractor couple in probabilistic normed spaces and discuss the solution for nonlinear equations of fuzzy mappings and the convergence of sequences generated by the algorithms in Menger probabilistic norm
Data Loading...
